Average Error: 32.9 → 12.5
Time: 1.6m
Precision: 64
Internal Precision: 3392
\[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.1201934423675601 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{-2}{3} \cdot b}{a}\\ \mathbf{elif}\;b \le 3.200932438162374 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}{3}}{a}\\ \mathbf{elif}\;b \le 6.804319749141088 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(3 \cdot a\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{c \cdot a}{b} \cdot \frac{-3}{2}}{3}}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus c

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if b < -1.1201934423675601e+112

    1. Initial program 46.5

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*46.5

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    4. Taylor expanded around -inf 3.9

      \[\leadsto \frac{\color{blue}{\frac{-2}{3} \cdot b}}{a}\]

    if -1.1201934423675601e+112 < b < 3.200932438162374e-57

    1. Initial program 13.2

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*13.3

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    if 3.200932438162374e-57 < b < 6.804319749141088e+153

    1. Initial program 44.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied flip-+44.1

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a}\]

    4. Applied associate-/l/45.6

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}}\]

    5. Simplified15.6

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) + c \cdot \left(a \cdot 3\right)}}{\left(3 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right)}\]

    if 6.804319749141088e+153 < b

    1. Initial program 62.8

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}\]
    2. Using strategy rm

    3. Applied associate-/r*62.8

      \[\leadsto \color{blue}{\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3}}{a}}\]

    4. Taylor expanded around inf 14.3

      \[\leadsto \frac{\frac{\color{blue}{\frac{-3}{2} \cdot \frac{c \cdot a}{b}}}{3}}{a}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.1201934423675601 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{-2}{3} \cdot b}{a}\\ \mathbf{elif}\;b \le 3.200932438162374 \cdot 10^{-57}:\\ \;\;\;\;\frac{\frac{\left(-b\right) + \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}}{3}}{a}\\ \mathbf{elif}\;b \le 6.804319749141088 \cdot 10^{+153}:\\ \;\;\;\;\frac{\left(b \cdot b - b \cdot b\right) + c \cdot \left(3 \cdot a\right)}{\left(\left(-b\right) - \sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)}\right) \cdot \left(3 \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{c \cdot a}{b} \cdot \frac{-3}{2}}{3}}{a}\\ \end{array}\]

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018214 
(FPCore (a b c d)
  :name "Cubic critical"
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3 a) c)))) (* 3 a)))